Find the maxima, minima, and saddle points of the function f(x,y) = x4 + y4 + 4xy.

You're on a roll — To find the maxima, minima, and saddle points of the function f(x,y) = x^4 + y^4 + 4xy, we follow these steps: Step 1: Find the first partial derivatives and set them to zero to find critical points. The given function is f(x,y) = x^4 + y^4 + 4xy. The partial derivative with respect to x is: f_x = ()/( x)(x^4 + y^4 + 4xy) = 4x^3 + 4y The partial derivative with respect to y is: f_y = ()/( y)(x^4 + y^4 + 4xy) = 4y^3 + 4x Set f_x = 0 and f_y = 0: 1) 4x^3 + 4y = 0 x^3 + y = 0 y = -x^3 2) 4y^3 + 4x = 0 y^3 + x = 0 Substitute y = -x^3 from equation (1) into equation (2): (-x^3)^3 + x = 0 -x^9 + x = 0 x(1 - x^8) = 0 This equation yields three possible values for x: x = 0 1 - x^8 = 0 x^8 = 1 x = ± 1 Now, find the corresponding y values using y = -x^3: If x = 0, then y = -(0)^3 = 0. Critical point: (0,0). If x = 1, then y = -(1)^3 = -1. Critical point: (1,-1). If x = -1, then y = -(-1)^3 = 1. Critical point: (-1,1). The critical points are (0,0), (1,-1), and (-1,1). Step 2: Find the second partial derivatives. f_xx = ()/( x)(4x^3 + 4y) = 12x^2 f_yy = ()/( y)(4y^3 + 4x) = 12y^2 f_xy = ()/( y)(4x^3 + 4y) = 4 Step 3: Calculate the discriminant D(x,y) = f_xxf_yy - (f_xy)^2. D(x,y) = (12x^2)(12y^2) - (4)^2 = 144x^2y^2 - 16 Step 4: Apply the Second Derivative Test to each critical point. For the critical point (0,0): D(0,0) = 144(0)^2(0)^2 - 16 = -16 Since D(0,0) < 0, the point (0,0) is a saddle point. For the critical point (1,-1): D(1,-1) = 144(1)^2(-1)^2 - 16 = 144 - 16 = 128 Since D(1,-1) > 0, we check f_xx: f_xx(1,-1) = 12(1)^2 = 12 Since f_xx(1,-1) > 0, the point (1,-1) is a local minimum. The function value at this point is: f(1,-1) = (1)^4 + (-1)^4 + 4(1)(-1) = 1 + 1 - 4 = -2 For the critical point (-1,1): D(-1,1) = 144(-1)^2(1)^2 - 16 = 144 - 16 = 128 Since D(-1,1) > 0, we check f_xx: f_xx(-1,1) = 12(-1)^2 = 12 Since f_xx(-1,1) > 0, the point (-1,1) is a local minimum. The function value at this point is: f(-1,1) = (-1)^4 + (1)^4 + 4(-1)(1) = 1 + 1 - 4 = -2 The final answer is: Saddle point at (0,0) Local minimum at (1,-1) with value -2 Local minimum at (-1,1) with value -2 Send me the next one 📸